Hundreds of American women, most of them welfare recipients from Brooklyn and the Bronx, suddenly seemed to be finding romance while vacationing in Bangladesh's main port city.

Alas, it wasn't love, but money, that led them into whirlwind marriages with Bangladeshi men eager to seek their fortunes in the United States.

The "grooms," mostly from humble origins in Chittigong or nearby Sandwip Island, would pay up to $35,000 each to a criminal ring based in New York that would arrange the sham nuptials and file petitions for the men to receive immigrant visas as spouses of U.S. citizens. The "brides" would get up to $2,000 each and a free trip to Bangladesh out of the deal.

In what one State Department report called a "grotesque effort" to establish the marriages' conjugal bona fides, the applications typically would contain a snapshot of the happy couple, usually in an advanced state of undress, embracing on a hotel bed during their "honeymoon."

The "Brooklyn brides scam," as investigators dubbed it, is just one example of the thousands of f rauds that are employed every day to beat the U.S. immigration system.

It is a system that U.S. officials say is under daily attack from con artists around the globe. While most travel to this country is legitimate, visa fraud has become a growing problem for U.S. diplomatic missions, immigration inspectors and local governments.

"It's a never ending battle," said Mary A. Ryan, assistant secretary of state for consular affairs. "Fraud is a problem almost everywhere for us."

Many illegal immigrants obtain legitimate visas through fraudulent means. Others come in with bogus documents, or no documents. And thousands get to the United States on temporary visas for students or tourists and then stay on after the visas expire.

"It's easy to think of illegal immigration as poor souls coming across the border," Ryan said. "But every day planes land in this country with people who have fraudulent documents or people who have destroyed their documents en route. That's what we have to combat."

U.S. authorities estimate that as many as 5 million illegal immigrants reside in the United States on a long-term basis and that this population increases at a net rate of 300,000 to 400,000 a year. Using incomplete and disputed data, U.S. officials figure that anywhere from a third to half of this population came in through legal ports of entry while the rest slipped across the border.

Among the great many people who enter the country illegally to find work or to live with relatives, officials say, are some dangerous characters. "Every transnational criminal organization relies on fraudulent documents," said Jonathon Winer, deputy assistant secretary of state for international narcotics and law enforcement affairs. "Getting a visa is a critical element to be able to carry out crime across borders."

Immigration and Naturalization Service (INS) inspectors intercepted 170,000 fraudulent documents in 1993 at ports of entry alone, according to officials at the INS Forensic Document Laboratory, an immigration crime lab that claims to have the world's biggest collection of travel documents. The take routinely includes counterfeit passports, fake visas, passports with the photo or visa page substituted, false alien registration cards and legitimate documents used by impostors.

The visa scams range from clumsy charades at U.S. consular offices to sophisticated ploys that prove difficult to stop. No one knows how much fraud goes undetected.

"Look-alike" frauds, in which the impostor closely resembles the person pictured in a legal document, have proliferated in recent years as new immigrant communities have burgeoned. Alien smuggling gangs buy or steal documents from the expanding pool of new U.S. citizens or permanent residents, then reuse the documents to smuggle people into the country.

Faced with alien smugglers and illegal immigrants who exploit every visa category and loophole they can find, State Department officials say many consular offices have been swamped with more fraudulent applications than they can adequately investigate. Even obvious fraud attempts tax the offices' time and resources, they said, and the squeeze appears likely to intensify as the State Department braces for budget cuts of up to 30 percent.

The INS, an arm of the Justice Department, has seen its budget grow quickly in the past two years, but the lion's share of the new money has gone to increased enforcement along the U.S.-Mexico border. Substantial investments also have gone into new technology to keep better track of legal immigrants and visitors that should eventually help the agency fight fraud. Immigration bills under consideration in Congress include measures aimed at expediting deportations and countering illegal immigrants' use of fraudulent documents for U.S. employment. However, Congress has rejected INS requests for $22.5 million in fiscal 1996 for 106 new agents to support its counterfraud program and 98 new positions to fight alien smuggling.

The first line of defense, however, remains the consular posts overseas that issue most visas. "Our posts are frequently overwhelmed with fraud," said Ryan of the State Department's consular section. "We don't have enough resources, we don't have enough personnel to deal with this problem properly."

In some South Asian countries, U.S. consular officials estimate that 90 percent of the documents submitted to support visa applications are fake.

Russian mafia organizations have become adept at setting up phony companies and exploiting U.S. "L" visas for "intracompany transferees," investigators say.

In southern China, a recent survey by the U.S. Consulate in Guangzhou found that 80 percent of the companies listed in "L" visa applications did not exist. Thousands of Chinese also have taken advantage of U.S. political asylum laws to settle in America, even if they flew in with false documents or handed them off to smugglers on the plane.

In the "Brooklyn brides" case, investigators estimate that the ring had been operating since the early 1990s and that others like it had arranged fraudulent marriages since the early 1980s. All told, officials say, at least 1,000 Bangladeshi men had entered the United States through the scams. The investigators say the grooms' extended families would pool their resources, often mortgaging property, to pay the ring's fees in hopes of eventually emigrating themselves. Many later did so once the grooms became eligible to sponsor them.

In a similar case in Virginia, a Vietnamese-born immigration lawyer in Falls Church pleaded guilty in June to arranging more than 30 sham marriages, most of them between low-income area residents and citizens of the West African state of Sierra Leone, so that the Africans could obtain legal permanent resident status here. The lawyer, My-Linh Duong Soland, 54, charged the immigrants $5,000 each for the marriages and paid their U.S. partners several hundred dollars apiece, court documents Department report said. But the INS, lacking proof of fraud, instead revalidated them, and the embassy "had no choice but to issue" the visas, it said. Once the INS approves an immigrant petition, it is reluctant to revoke it for fear of being sued, consular officials explained.

Solid evidence of the fraud finally emerged in December 1993 when one of the "brides" was caught at Miami airport trying to smuggle in cocaine from Guyana. Under questioning by federal agents, the woman revealed details of the marriage scam and agreed to become an informer.

A six-month investigation disclosed that the ring was run by several Bangladeshi ship jumpers, who had become legal permanent U.S. residents, and at least one former "groom."

Last August, federal agents in New York arrested two of the Bangladeshis and an American immigration lawyer in the case. Although the U.S. Embassy in Dhaka had already begun rejecting visas for "grooms" by that time, it has continued to receive several applications a month, and diplomatic cables on the subject have taken on a sarcastic tone.

"Lovelorn males from Sandwip continue to apply for {immigrant visas} to join their loved ones in Brooklyn," one Oct. 17 cable reported. "The lovebirds meet in the tourist Mecca of Chittagong, fall in love, marry, honeymoon . . . and tearfully part ways, all in the course of less than a week." Furthermore, the cable complained, the honeymoon snapshots "have become boringly repetitive."

"We're aware of the reports it's still going on," a U.S. investigator said. "It doesn't shock us." He said the embassy was probably seeing the results of "various offshoots" of the main scam.

As one of the poorest and most densely populated countries in the world, Bangladesh provides U.S. consular officials with a daily diet of visa fraud attempts. Many are fairly transparent. One application included a business letter from someone named "Mike" in Indiana that ended with the plea, "Begging your honour to issue the necessary visa, please do the needful and thereby oblige."

In a more elaborate scheme, a Bangladeshi emigre recently attempted to obtain visas for a 31-member "basketball team" to compete against U.S. college teams in the Northwest. Although the applicant presented a legitimate invitation from a Seattle-based organization, several other aspects of the case "stretched credulity," the U.S. Embassy in Dhaka reported.

For one thing, of the 24 players listed, only six were over 5 feet 7 inches tall -- "rather short even by Bangladeshi standards," the embassy noted in a cable -- and none was over 5 foot 10. Seven "coaches and managers" on the roster were all between 25 and 30 years old. The Bangladeshi organizer, who described himself as a banker and basketball fan with a residence in New York, was unable to tell a consular officer the height of the basketball rim, the location of the free throw lane or the

Investigations by the INS and the State Department's Bureau of Diplomatic Security occasionally reveal corruption in U.S. embassies abroad, usually committed by locally hired employees.

In a recent case on the Caribbean island of Barbados, a local U.S. Embassy employee admitted selling more than 100 visas to a Syrian broker for $2,000 each.

For elaborate preparation and sheer chutzpah, it is probably hard to top a man from mainland China who gave his name as Zhang Xin when he presented a Venezuelan passport and requested a visitor's visa at the U.S. consulate in Istanbul in September. The passport was listed in an intelligence report as one of several thousand missing blank Venezuelan passports that were believed to have been sold by corrupt officials.

Zhang also presented what a State Department cable described as "an impressive array of documents," including a valid Turkish residence permit, a letter from medical supplies company in which he was a partner, company tax and registration documents and a bank book showing a balance of nearly $60,000.

"Appearing poised and confident, and speaking in both English and Turkish, Mr. Zhang explained that he obtained his Venezuelan passport by marrying a Venezuelan citizen several years ago," the cable reported. But Zhang said he could not speak Spanish because he had divorced his wife after only two years of marriage in Venezuela.

The consular officer nevertheless pressed him on this point, asking him to search his memory for a single word of Spanish.

"After thinking long and hard," the cable said, "Mr. Zhang finally came up with ciao.' "

And that, essentially, was what the embassy and the Turkish government said to Zhang. After learning that his "legal" status in Turkey was entirely based on his false Venezuelan citizenship, Turkish authorities ordered him deported